Munich Personal RePEc Archive
Fixed point theorems for multivalued contractive operators on generalized metric spaces
Guran, Liliana
"Titu Maiorescu" University of Bucharest, Romania
26 October 2010
Online at https://mpra.ub.uni-muenchen.de/26204/
MPRA Paper No. 26204, posted 01 Nov 2010 00:33 UTC
Fixed point theorems for multivalued contractive operators on generalized
metric spaces
Liliana Guran
Department of Accounting and Managerial Information System, Faculty of Economic Sciences, Titu Maiorescu University Dˆambovnicului 22, 040441, sector 4, Bucharest, Romania.
E-mail:gliliana.math@gmail.com
Abstract
In this paper we give a fixed point results for multivalued operators on generalized metric spaces endowed with a generalizedw-distance. Then we study the data dependence for this new result.
Key words and phrases: multivalued weakly Picard operator, w-distance, fixed point, multivalued operator.
Mathematics Subject Classification 2000: 47H10, 54H25.
1 Introduction
It is well known that Caristis fixed point theorem [2] is equivalent to Ekland variational principle [4], which is nowadays is an important tool in nonlin- ear analysis. Most recently, many authors studied and generalized Caristis fixed point theorem to various directions. Using the concept of Hausdorff met- ric, Nadler Jr. [13] has proved multivalued version of the Banach contraction
principle which states that each closed bounded valued contraction map on a complete metric space, has a fixed point.
Recently, Bae [1] introduced a notion of multivalued weakly contractive maps and applying generalized Caristis fixed point theorems he proved several fixed point results for such maps in the setting of metric and Banach spaces.
Many authors have been using the Hausdorff metric to obtain fixed point results for multivalued maps on metric spaces, but, in fact for most cases the existence part of the results can be proved without using the concept of Hausdorff metric.
Recently, using the concept of w-distance [9], Suzuki and Takahashi [20]
introduced a notion of multivalued weakly contractive in short, w-contractive maps and improved the Nadlers fixed point result without using the concept of Hausdorff metric. Most recently, Latif [10] generalized the fixed point result of Suzuki and Takahashi [[20], Theorem 1].
The concept of multivalued weakly Picard operator (briefly MWP oper- ator) was introduced in close connection with the successive approximation method and the data dependence phenomenon for the fixed point set of mul- tivalued operators on complete metric space, by I. A. Rus, A. Petru¸sel and A.
Sˆant˘am˘arian, see [19]. In [17] is presented the theory of multivalued weakly Picard operators in L-spaces.
In 1966 A. I. Perov was introduced the concept of generalized metric space and obtained a generalization of the Banach principle for contractive operators on spaces endowed with vector-valued metrics, see [16].
The purpose of this paper is to recall the notion of generalized w-distance in a generalized metric space. Also, we present some generalizations of some fixed point results obtained in [5] with respect to a generalized w-distance and we give a data dependence result for the new theorem of fixed point.
2 Preliminaries
Let (X, d) be a metric space. We will use the following notations:
P(X) -the set of all nonempty subsets of X; P(X) =P(X)S
∅
Pcl(X) - the set of all nonempty closed subsets of X; Pb(X) - the set of all nonempty bounded subsets of X;
Pb,cl(X) - the set of all nonempty bounded and closed, subsets of X; For two subsets A, B ∈Pb(X) we recall the following functionals.
δ : P(X) × P(X) → R+, δ(A, B) := sup{d(a, b)|x ∈ A, b ∈ B} - the diameter functional;
H : P(X)× P(X) → R+, H(A, B) :=max{sup
a∈A
b∈Binfd(a, b),sup
b∈B
a∈Ainf d(a, b)}
-the Pompeiu-Hausdorff functional;
If T :X → P(X) is a multivalued operator, then we denote by F ixT the fixed point set ofT, i.e. F ixT :={x∈X|x∈T(x)}.
First we define the concept of L-space.
Definition 2.1 Let X be a nonempty set and s(X) := {(xn)n∈N|xn ∈ X, n ∈ N}. Let c(X) ⊂ s(X) a subset of s(X) and Lim : c(X) → X an operator.
By definition the triple (X, c(X),Lim) is called an L-space if the following conditions are satisfied:
(i) If xn=x, for alln ∈N, then (xn)n∈N∈c(X) and Lim(xn)n∈N=x.
(ii) If (xn)n∈N ∈ c(X) and Lim(xn)n∈N = x, then for all subsequences, (xni)i∈N, of (xn)n∈N we have that (xni)i∈N∈c(X) and Lim(xni)i∈N =x.
By the definition an element ofc(X) is convergent and x:=Lim(xn)n∈N is the limit of this sequence and we can write xn→x as n→ ∞.
We will denote an L-space by (X,→).
Let us give some examples of L-spaces, see [17].
Example 2.1 (L-structures on Banach spaces) Let X be a Banach space. We denote by → the strong convergence in X and by ⇀ the weak convergence in X. Then (X,→), (X, ⇀) are L-spaces.
Example 2.2 (L-structures on function spaces) let X and Y be two metric spaces. Let M(X, Y) the set of all operators from X to Y. We denote by →p
the point convergence on M(X, Y), by unif→ the uniform convergence and by
cont→ the convergence with continuity. Then (M(X, Y),→),p (M(X, Y),unif→) and (M(X, Y),cont→) are L-spaces.
Definition 2.2 Let (X,→) be an L-space. Then T :X → P(X) is a multi- valued weakly Picard operator(briefly MWP operator)if for each x ∈ X and each y∈T(x) there exists a sequence (xn)n∈N in X such that:
(i)x0 =x, x1 =y;
(ii)xn+1 ∈T(xn), for alln ∈N;
(iii)the sequence (xn)n∈N is convergent and its limit is a fixed point of T. Let us give some examples of MWP operators, see [17],[19].
Example 2.3 Let (X, d) be a complete metric space and T :X → Pcl(X) be a Reich type multivalued operator, i.e. there exists α, β, γ ∈R+ with
α+β+γ <1 such that
H(T(x), T(y))≤αd(x, y) +βD(x, T(x)) +γD(y, T(y)), for all x, y ∈X. Then T is a MWP operator.
Example 2.4 Let (X, d) be a complete metric space andT :X →Pcl(X)be a closed multifunction for which there existsα, β ∈R+ withα+β <1 such that H(T(x), T(y))≤αd(x, y) +βD(y, T(y)),for everyx∈X and everyy∈T(x).
Then T is a MWP operator.
Example 2.5 Let (X, d) be a complete metric space and T1, T2 :X →Pcl(X) for which there exists α∈]0,12[ such that
H(T1(x), T2(y))≤α[D(x, T1(x)) +D(y, T2(y))], for each x, y ∈X. Then T1 and T2 are a MWP operators.
The concept of w-distance was introduced by O. Kada, T. Suzuki and W. Takahashi (see[9]) as follows:
Let (X,d) be a metric space. A functional w : X×X → [0,∞) is called w-distance on X if the following axioms are satisfied :
1. w(x, z)≤w(x, y) +w(y, z), for any x, y, z ∈X;
2. for any x∈X :w(x,·) :X →[0,∞) is lower semicontinuous;
3. for any ε >0, exists δ >0 such that w(z, x)≤δ and w(z, y)≤δ implies d(x, y)≤ε.
Some examples of w-distance are given in [9].
Example 2.6 Let (X, d) be a metric space. Then the metric ”d” is a w- distance on X.
Example 2.7 Let X be a normed linear space with norm || · ||. Then the function w : X × X → [0,∞) defined by w(x, y) = ||x|| + ||y|| for every x, y ∈X is a w-distance.
Example 2.8 Let (X, d) be a metric space and let g : X → X a continuous mapping. Then the function w:X×Y →[0,∞) defined by:
w(x, y) =max{d(g(x), y), d(g(x), g(y))}
for every x, y ∈X is a w-distance.
Let us recall a crucial lemma for w-distance (see[20] for more details).
Lemma 2.1 Let (X, d) be a metric space, and let w be a w-distance on X.
Let(xn)and(yn)be two sequences inX, let (αn),(βn)be sequences in[0,+∞[
converging to zero and let x, y, z ∈X. Then the following statements hold:
1. If w(xn, y)≤αn and w(xn, z)≤βn for any n ∈N, then y=z.
2. If w(xn, yn)≤αn and w(xn, z)≤ βn for any n ∈N, then (yn) converges to z.
3. If w(xn, xm)≤ αn for any n, m ∈N with m > n, then (xn) is a Cauchy sequence.
4. If w(y, xn)≤αn for any n∈N, then (xn) is a Cauchy sequence.
For the rest of the paper, if v, r∈Rm, v := (v1, v2,· · · , vm) and
r := (r1, r2,· · · , rm), then by v ≤ r means vi ≤ri, for each i∈ {1,2,· · · , m}, while v < r meansvi < ri, for each i∈ {1,2,· · · , m}.
Also, |v|:= (|v1|,|v2|,· · · ,|vm|) and, if c∈R then v ≤cmeans vi ≤ci, for eachi∈ {1,2,· · · , m}.
Ifx0 ∈X and r∈Rm
+ withri >0 for eachi∈ {1,2,· · · , m}we will denote byB(x0;r) :={x∈ X|d(x0, x)< r}the open ball centered in x0 with radius r := (r1, r2,· · · , rm) and by Be(x0;r) :={x ∈X| d(x0, x) ≤r} the closed ball centered in x0 with radius r.
In [8] we can find the notion of generalized w-distance as follows.
Definition 2.3 Let (X, d) a generalized metric space. The mapping e
w : X×X → Rm
+ defined by w(x, y) = (ve 1(x, y), v2(x, y), ..., vm(x, y)) is said to be a generalized w-distance if it satisfies the following conditions:
(w1) w(x, y)e ≤w(x, z) +e w(z, y), for everye x, y, z ∈X;
(w2) vi :X×X →R+ is lower semicontinuous, for i∈ {1,2, . . . , m};
(w3) For any ε:= (ε1, ε2, ..., εm)>0, for m∈N, there exists δ:= (δ1, δ2, ..., δm)>0 such that w(z, x)e ≤δ and w(z, y)e ≤δ implies d(x, y)e ≤ε.
The notion of generalized w-distance with his properties was discussed in [8].
Let us present now an important lemma for w-distances into the terms of generalized w-distances.
Lemma 2.2 Let(X,d)e be a generalized metric space, and letwe:X×X →Rm
+
be a generalized w-distance on X. Let (xn) and (yn) be two sequences in X, let αn = (α(1)n , α(2)n , ..., α(m)n ) ∈ R+ and βn = (βn(1), βn(2), ..., βn(m)) ∈ R+ be two sequences such that α(i)n and βn(i) converge to zero for each i ∈ {1,2, . . . , m}.
Let x, y, z ∈X. Then the following assertions hold:
1. If w(xe n, y)≤αn and w(xe n, z)≤βn for any n ∈N, then y=z.
2. If w(xe n, yn)≤αn and w(xe n, z)≤ βn for any n ∈N, then (yn) converges to z.
3. If w(xe n, xm)≤ αn for any n, m ∈N with m > n, then (xn) is a Cauchy sequence.
4. If w(y, xe n)≤αn for any n∈N, then (xn) is a Cauchy sequence.
Throughout this paper we will denote by Mm,m(R+) the set of all m×m matrices with positive elements, by Θ the zerom×mmatrix, by I the identity m×m matrix and by U the unity m×m matrix. If A∈Mm,m(R+), then the symbol Aτ stands for the transpose matrix ofA.
Recall that a matrixAis said to be convergent to zero if and only ifAn→0 asn → ∞.
For the proof of the main result we need the following theorem, see [16].
Theorem 2.1 Let A∈Mm,m(R+). The following statements are equivalent:
(i) A is a matrix convergent to zero;
(i) An →0 as n → ∞;
(ii)The eigen-values of Aare in the open unit disc, i.e. |λ|<1, for every λ∈C with det(A−λI) = 0;
(iii) The matrix I−A is non-singular and
(I−A)−1 =I +A+...+An+...;
(iv) The matrix I−A is non-singular and (I −A)−1 has nonnenegative elements.
(v) Anq →0 and qAn →0 as n → ∞, for each q∈Rm.
3 Main results
Throughout this section (X, d) is a generalized metric space in Perov’s sense and w is a generalized w-distance on the generalized metric space.
Let x0 ∈X and r:= (ri)ni=1 for each i={1,2, ..., m}. Let us define:
Bew(x0;r) := {x ∈ X|w(xe 0, x) < r} the open ball centered at x0 with radius r with respect tow;e
Bew(x0;r) := {x ∈ X|w(xe 0, x) ≤ r} the closed ball centered at x0 with radius r with respect tow;e
Bewd(x0;r)- the closure in (X, d) of the set Bew(x0;r).
Theorem 3.2 Let (X, d) be a complete generalized metric space, x0 ∈ X, r := (ri)ni=1 for each i = {1,2, ..., m}, we : X × X → [0,∞) a generalized w-distance on X and let T : Bew(x0;r) → P(X) be a multivalued operator with the property that there exists A= (ai,j)i,j∈{1,2,...,m} ∈ Mm,m(R+) a matrix convergent to zero and B = (bi,j)i,j∈{1,2,...,m} ∈ Mm,m(R+)\ {U} such that, for every x, y ∈X and each u∈T(x), there exists v ∈T(y) such that
e
w(u, v)≤Aw(x, y) +e BDwe(y, T(y)), where Dwe(x, T(x)) :=inf{w(x, y) :e y∈T(x)}.
(This means, that for each x, y ∈ Y and each u ∈ T(x), there exists v ∈T(y) such that
w1(u, v)
· · · wm(u, v)
≤
a11 · · · a1,m
· · ·
am1 · · · am,m
·
w1(x, y)
· · · wm(x, y)
+
b11 · · · b1,m
· · ·
bm1 · · · bm,m
·
Dw1(x, y)
· · · Dwm(x, y)
)
Suppose that:
1. inf{w(x, y) +e Dwe(x, T(x))}>0, for every x, y ∈X and y /∈T(y).
2. There exists x1 ∈T(x0) such that w(xe 0, x1)(I−A)−1 ≤r.
3. If u∈Rm
+ is such that u(I−A)−1 ≤(I −A)−1r, then u≤r.
Then F ixT 6=∅.
Proof. Letx0 ∈X and x1 ∈T(x0) such that e
w(x0, x1)(I−A)−1 ≤r≤(I−A)−1·r
Then, by (2),x1 ∈Bew(x0;r). Forx1 ∈T(x0) there existsx2 ∈T(x1) such that e
w(x1, x2)≤Aw(xe 0, x1) +BDwe(x1, T(x1))
≤Aw(xe 0, x1) +Bw(xe 1, x2) Thus w(xe 1, x2)≤ U−BA w(xe 0, x1)
We denote C := U−BA and we observe that the matrix C ∈ Mm,m(R) is a matrix convergent to zero and satisfy the following inequalities
• I+U−BA ≤I+A+A2+· · ·+An+· · ·, thereforeI +C ≤(I−A)−1
• (I−C)−1 ≤(I−A)−1
Thus w(xe 1, x2)(I − A)−1 ≤ U−BA w(x0, x1)(I − A)−1 ≤ Cr. Notice that x2 ∈Bew(x0;r).
Indeed, since w(xe 0, x2) ≤ w(xe 0, x1) +w(xe 1, x2) we get that w(x0, x2)(I − A)−1 ≤w(xe 0, x1)(I−A)−1+w(xe 1, x2)(I−A)−1 ≤Ir+Cr≤(I−A)−1r, which immediately implies (by hypothesis (2)) thatw(xe 0, x2)≤r.
By induction, we construct the sequence (xn)n∈N in Bfw(x0;r) having the properties:
(a) xn+1 ∈T(xn), n ∈N;
(b) w(xe 0, xn)(I − A)−1 ≤ (I − A)−1r, for each n ∈ N∗, that means e
w(x0, xn)≤r;
(c) w(xe n, xn+1)(I−A)−1 ≤Cnr, for each n ∈N. By (c), for every m, n∈N, withm > n, we get that
e
w(xn, xm)(I−A)−1 ≤Cn(I−C)−1r ≤Cn(I−A)−1r.
By Lemma 2.2(3) we have that the sequence (xn)n∈N is Cauchy in the complete metric space. Denote by x∗ its limit in Bewd(x0;r).
Assume that x∗ ∈/ T(x∗). Fix n ∈ N. Since (xm)m∈N is a sequence in Bew(x0;s) which converge to x∗ and w(xe n,·) is lower semicontinuous we have
e
w(xn, x∗)≤ lim
m→∞infw(xe n, xm)≤Cnr, for every n∈N.
Therefore, by hypothesis (1) and using above inequality we have
0≤inf{w(x, xe ∗) +Dwe(x, T(x)) :x∈X}
≤inf{w(xe n, x∗) +w(xe n, xn+1) :n∈N}
≤inf{2Cnr}= 0.
Which is a contradiction. Thus conclude that x∗ ∈T(x∗).
A global version of the previous theorem is the following result.
Theorem 3.3 Let (X, d) be a complete generalized metric space, x0 ∈ X, r := (ri)ni=1 for each i = {1,2, ..., m}, we : X × X → [0,∞) a generalized w-distance on X and let T : X → P(X) be a multivalued operator with the property that there exists A ∈ Mm,m(R+) a matrix convergent to zero and B ∈ Mm,m(R+)\ {U} such that, for every x, y ∈X and each u ∈T(x), there exists v ∈T(y) such that
e
w(u, v)≤Aw(x, y) +e BDwe(y, T(y)), where Dwe(x, T(x)) :=inf{w(x, y) :e y∈T(x)}.
Suppose that inf{w(x, y) +e Dwe(x, T(x))} > 0, for every x, y ∈ X and y /∈ T(y) then
1. F ixT 6=∅.
2. The sequence (xn)n∈N∈X given by relation xn+1 ∈T(xn), for alln ∈N, is convergent and its limit is a fixed point of T.
3. One has the estimation w(xe n, x∗) ≤ Cnw(xe 0, x1) where C ∈ Mm,m(R), C := U−BA , and x∗ ∈F ixT.
Remark 3.1 In the condition of the previous theorem we observe that T is a MWP operator.
4 Data dependence theorem for weakly contractive type operators in generalized metric spaces
The main result of this section is the following data dependence theorem with respect to the Theorem 3.3.
Theorem 4.4 Let (X, d) be a complete generalized metric space, x0 ∈ X, r := (ri)ni=1 for each i = {1,2, ..., m}, we : X × X → [0,∞) a generalized w-distance on X and let T1, T2 : X → P(X) be a multivalued operator with the property that there exists A∈ Mm,m(R+) a matrix convergent to zero and B ∈ Mm,m(R+)\ {U} such that, for every x, y ∈ X and each u ∈ Tj(x), for every j ∈ {1,2}, there exists v ∈Tj(y) such that
e
w(u, v)≤Aw(x, y) +e BDwe(y, Tj(y)), where Dwe(x, Tj(x)) := inf{w(x, ye ) :y∈Tj(x)}.
Suppose that the following are true:
1. F ixT1 6=∅ 6=F ixT2.
2. We suppose that there exists η := (ηi)ni=1, for each i={1,2, ..., m}, with η > 0, such that for every u ∈ T1(x) there exists v ∈ T2(x) such that
e
w(u, v)≤η, (respectively for everyv ∈T2(x) there exists u∈T1(x) such that w(v, u)e ≤η).
3. inf{w(x, y) +e Dwe(x, Tj(x))} > 0 for each j ∈ {1,2}, for every x, y ∈ X and y /∈Tj(y).
Then for every u∗ ∈F ixT1 there exists v∗ ∈F ixT2 such that e
w(u∗, v∗)≤U(1−C)−1η, where C ∈ Mm,m(R), C:= U−BA ; (respectively for every v∗ ∈F ixT2 there exists u∗ ∈F ixT1 such that
e
w(v∗, u∗)≤U(1−C)−1η, where C ∈ Mm,m(R), C := U−BA ).
Proof.Letu0 ∈F ixT1, thenu0 ∈T1(u0). Using the hypothesis (2) we have that there existsu1 ∈T2(u0) such that w(ue 0, u1)≤η.
We have that for everyu0, u1 ∈X withu1 ∈T2(u0) there existsu2 ∈T2(u1) such thatw(ue 1, u2)≤Aw(ue 0, u1)+BDwe(u1, T2(u1))≤Aw(ue 0, u1)+Bw(ue 1, u2).
Thus w(ue 1, u2)≤ U−BA w(ue 0, u1)
We denote C := U−BA and we observe that the matrix C ∈ Mm,m(R) is a matrix convergent to zero.
Thus w(ue 1, u2)≤Cw(ue 0, u1).
By induction we obtain a sequence (un)n∈N∈X such that (1) un+1 ∈T2(un), for every n∈N;
(2) w(ue n, un+1)≤Cnw(ue 0, u1) For n, p∈N we have the inequality
e
w(un, un+p)≤Cn(I−C)−1w(ue 0, u1).
By the Lemma 2.2(3) we have that the sequence (un)n∈N is a Cauchy se- quence. Since (X, d) is a complete metric space we have that there existsv∗ ∈X such thatun
→d v∗.
Assume that v∗ ∈/T2(v∗). Fix n∈N. By the lower semicontinuity of e
w(x,·) :X →[0,∞) we have e
w(un, v∗)≤ lim
p→∞infw(ue n, un+p)≤Cn(I−C)−1w(ue 0, u1) (4.1) Therefore, by hypothesis (3) and using the relation 4.1 we have the inequality:
0 <inf{w(u, ve ∗) +Dwe(u, T2(u)) :x∈X}
≤inf{w(ue n, v∗) +w(ue n, un+1) :n ∈N}
≤inf{Cn(I−C)−1w(ue 0, u1) +Cnw(ue 0, u1) :n ∈N}= 0.
Which is a contradiction. Thus we conclude that v∗ ∈T2(v∗).
Then, by w(ue n, v∗) ≤ Cn(I − C)−1w(ue 0, u1), with n ∈ N, for n = 0 we obtain w(ue 0, v∗) ≤ U(I−C)−1w(ue 0, u1) ≤ U(I−C)−1η, which complete the proof. ✷
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